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Some distributions in increasing and flattened permutations

Jean-Luc Baril, José L. Ramírez

Abstract

We examine the distribution and popularity of different parameters (such as the number of descents, runs, valleys, peaks, right-to-left minima, and more) on the sets of increasing and flattened permutations. For each parameter, we provide an exponential generating function for its corresponding distribution and popularity.Additionally, we present one-to-one correspondences between these permutations and some classes of simpler combinatorial objects.

Some distributions in increasing and flattened permutations

Abstract

We examine the distribution and popularity of different parameters (such as the number of descents, runs, valleys, peaks, right-to-left minima, and more) on the sets of increasing and flattened permutations. For each parameter, we provide an exponential generating function for its corresponding distribution and popularity.Additionally, we present one-to-one correspondences between these permutations and some classes of simpler combinatorial objects.

Paper Structure

This paper contains 11 sections, 25 theorems, 89 equations, 1 figure, 1 table.

Table of Contents

  1. Introduction and Notation
  2. Enumeration of Increasing Permutations
  3. For the number of valleys: ${\textsf{s}}={\textsf{val}}$
  4. For the number of runs: s=run=1+des=n-asc
  5. For the number of right-to-left minima: s=rlm
  6. For the number of peaks: s=peak
  7. Alternating increasing permutations
  8. Flattened Permutations
  9. For the number of valleys: ${\textsf{s}}={\textsf{val}}$
  10. For the number of runs: ${\textsf{s}}={\textsf{run}}=1+{\textsf{des}}=1+{\textsf{peak}}=n-{\textsf{asc}}$
  11. For the number of right-to-left minima: ${\textsf{s}}={\textsf{rlm}}$

Key Result

Theorem 2.1

The generating function for increasing permutations with respect to the length and the number of valleys is

Figures (1)

  • Figure 1: Increasing permutation $\pi=6~1~2~8~4~7~5~9~3$, and flattened permutation $\pi=1~5~2~4~3~6~9~7~8$.

Theorems & Definitions (35)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • Corollary 2.3
  • Corollary 2.4
  • Theorem 2.5
  • proof
  • Corollary 2.6
  • Corollary 2.7
  • Theorem 2.8
  • ...and 25 more